express
Dynamicallytyped expressions involving function application and variables.
https://github.com/rudymatela/express#readme
LTS Haskell 19.25:  1.0.10 
Stackage Nightly 20220925:  1.0.10 
Latest on Hackage:  1.0.10 
express1.0.10@sha256:e9ee81a2d5103721019f98d0faff29a3a01c658577e0bd4b19d8772ce2c5b881,11064
Module documentation for 1.0.10
Express
Express is a library for manipulating dynamically typed Haskell expressions.
It’s like Data.Dynamic
but with support for encoding applications and
variables.
It provides the Expr
type and over a hundred functions for
building, evaluating, comparing, folding, canonicalizing and matching
Expr
s. See Express’s Haddock documentation for more details.
This library has been used in the implementation of Speculate and Extrapolate.
Installing
To install the latest Express version from Hackage, just run:
$ cabal update
$ cabal install express
Starting from Cabal v3.0, you need to pass lib
as an argument to cabal
install:
$ cabal install express lib
Basics
To import Express
just:
> import Data.Express
For types that are Show
instances,
we can use val
to encode values as Expr
s.
> let false = val False
> :t false
false :: Expr
> print false
False :: Bool
> let one = val (1 :: Int)
> :t one
one :: Expr
> print one
1 :: Int
As seen above, the Show
instance for Expr
produces a string with the
encoded value and it’s type.
For types that aren’t Show
instances, like functions,
we can use value
to encode values as Expr
s.
> let notE = value "not" not
> :t notE
notE :: Expr
> print notE
not :: Bool > Bool
Using :$
we can apply function valued Expr
s, to other Exprs.
> let notFalse = notE :$ false
> :t notFalse
notFalse :: Expr
> notFalse
not False :: Bool
Using evaluate
and eval
we can evaluate Expr
s back into a regular Haskell value.
> evaluate notFalse :: Maybe Bool
Just True
> evaluate notFalse :: Maybe Int
Nothing
> eval False notFalse
True
> eval (0::Int) notFalse
0
Example 1: heterogeneous lists
Like with Data.Dynamic
, we can use Express to create heterogeneous lists.
Here, we use applications of val
to create a heterogeneous list:
> let xs = [val False, val True, val (1::Int), val (2::Int), val (3::Integer), val "123"]
> :t xs
xs :: [Expr]
> xs
[ False :: Bool
, True :: Bool
, 1 :: Int
, 2 :: Int
, 3 :: Integer
, "123" :: [Char]
]
We can then apply evaluate
to select values of different types:
> import Data.Maybe
> mapMaybe evaluate xs :: [Bool]
[False,True]
> mapMaybe evaluate xs :: [Int]
[1,2]
> mapMaybe evaluate xs :: [Integer]
[3]
> mapMaybe evaluate xs :: [String]
["123"]
Example 2: listing applications
Carrying on from Example 1, we define an heterogeneous list of functions
encoded as Expr
s:
> let fs = [value "not" not, value "&&" (&&), value "abs" (abs :: Int > Int)]
> :t fs
fs :: [Expr]
Using $$
we list the type correct applications of functions in fs
to
values in xs
.
> catMaybes [f $$ x  f < fs, x < xs]
[ not False :: Bool
, not True :: Bool
, (False &&) :: Bool > Bool
, (True &&) :: Bool > Bool
, abs 1 :: Int
, abs 2 :: Int
]
Example 3: uExtrapolate
uExtrapolate is a propertybased testing library capable of generalizing counterexamples. It’s implementation has under 40 lines of code. Besides, using Express to encode expressions, it uses LeanCheck for generating test values.
import Data.Express
import Test.LeanCheck hiding (counterExample, check)
Given a maximum number of tests and a property, the following counterExample
function returns either Nothing
when tests pass or Just
a counterexample
encoded as an Expr
.
counterExample :: (Listable a, Express a) => Int > (a > Bool) > Maybe Expr
counterExample maxTests prop = listToMaybe
[expr x  x < take maxTests list, not (prop x)]
Examples (REPL):
> counterExample 100 (\(x,y) > x + y == y + x)
Nothing
> counterExample 100 (\x > x == x + x)
Just (1 :: Integer)
> counterExample 100 (\xs > nub xs == (xs :: [Int]))
Just ([0,0] :: [Int])
Before moving on to generalize counterexamples, we need a way to compute ground
expressions from an expression with variables. For that, we will use grounds
and tiersFor
:
grounds :: Expr > [Expr]
grounds e = map (e //)
. concat
$ products [mapT ((,) v) (tiersFor v)  v < nubVars e]
tiersFor :: Expr > [[Expr]]
tiersFor e = case show (typ e) of
"Int" > mapT val (tiers :: [[Int]])
"Bool" > mapT val (tiers :: [[Bool]])
"[Int]" > mapT val (tiers :: [[ [Int] ]])
"[Bool]" > mapT val (tiers :: [[ [Bool] ]])
_ > []
Above, we restrict ourselves to Int
, Bool
, [Int]
and [Bool]
as test
types. So we can now compute the grounds of an expression with variables:
> grounds (value "not" not :$ var "p" (undefined :: Bool))
[ not False :: Bool
, not True :: Bool
]
> grounds (value "&&" (&&) :$ var "p" (undefined :: Bool) :$ var "q" (undefined :: Bool))
[ False && False :: Bool
, False && True :: Bool
, True && False :: Bool
, True && True :: Bool
]
To compute candidate generalizations from a given counterexample, we use the following function:
candidateGeneralizations :: Expr > [Expr]
candidateGeneralizations = map canonicalize
. concatMap canonicalVariations
. gen
where
gen e@(e1 :$ e2) =
[holeAsTypeOf e  isListable e]
++ [g1 :$ g2  g1 < gen e1, g2 < gen e2]
++ map (:$ e2) (gen e1)
++ map (e1 :$) (gen e2)
gen e
 isVar e = []
 otherwise = [holeAsTypeOf e  isListable e]
isListable = not . null . tiersFor
The need for isListable
above makes sure we only replace by variables what we
can enumerate. Our candidate generalizations are listed in nonincreasing
order of generality:
> candidateGeneralizations (value "not" not :$ val False)
[ p :: Bool
, not p :: Bool
]
Prelude> candidateGeneralizations (value "" () :$ val False :$ val True)
[ p :: Bool
, p  q :: Bool
, p  p :: Bool
, p  True :: Bool
, False  p :: Bool
]
For a given maximum number of tests, property and counterexample, the following function returns a counterexample generalization if one is found. It goes through the list of candidate generalizations and returns the first for which all tests fail.
counterExampleGeneralization :: Express a => Int > (a > Bool) > Expr > Maybe Expr
counterExampleGeneralization maxTests prop e = listToMaybe
[g  g < candidateGeneralizations e
, all (not . prop . evl) (take maxTests $ grounds g)]
We can finally define our check
function, that will test a property and
report a counterexample and a generalization when either are found.
check :: (Listable a, Express a) => (a > Bool) > IO ()
check prop = putStrLn $ case counterExample 500 prop of
Nothing > "+++ Tests passed.\n"
Just ce > "*** Falsified, counterexample: " ++ show ce
++ case counterExampleGeneralization 500 prop ce of
Nothing > ""
Just g > "\n generalization: " ++ show g
++ "\n"
Now we can find counterexamples and their generalizations:
> check $ \xs > sort (sort xs :: [Int]) == sort xs
+++ Tests passed.
> check $ \xs > length (nub xs :: [Int]) == length xs
*** Falsified, counterexample: [0,0] :: [Int]
generalization: x:x:xs :: [Int]
> check $ \x > x == x + (1 :: Int)
*** Falsified, counterexample: 0 :: Int
generalization: x :: Int
> check $ \(x,y) > x /= (y :: Int)
*** Falsified, counterexample: (0,0) :: (Int,Int)
generalization: (x,x) :: (Int,Int)
uExtrapolate has some limitations:
 it only supports properties with one argument (uncurried);
 it only supports generalization of
Int
,Bool
,[Int]
and[Bool]
values;  there is no way to configure the number of test arguments.
Please see Extrapolate for a fullfeatured version without the above limitations and with support for conditional generalizations.
Example 4: uSpeculate
Using Express, it takes less than 70 lines of code to define a function
speculateAbout
that conjectures equations about a set of functions based on
the results of testing:
> speculateAbout [hole (undefined :: Bool), val False, val True, value "not" not]
[ not False == True :: Bool
, not True == False :: Bool
, not (not p) == p :: Bool
]
> speculateAbout
> [ hole (undefined :: Int)
> , hole (undefined :: [Int])
> , val ([] :: [Int])
> , value ":" ((:) :: Int > [Int] > [Int])
> , value "++" ((++) :: [Int] > [Int] > [Int])
> , value "sort" (sort :: [Int] > [Int])
> ]
[ sort [] == [] :: Bool
, xs ++ [] == xs :: Bool
, [] ++ xs == xs :: Bool
, sort (sort xs) == sort xs :: Bool
, sort [x] == [x] :: Bool
, [x] ++ xs == x:xs :: Bool
, sort (xs ++ ys) == sort (ys ++ xs) :: Bool
, sort (x:sort xs) == sort (x:xs) :: Bool
, sort (xs ++ sort ys) == sort (xs ++ ys) :: Bool
, sort (sort xs ++ ys) == sort (xs ++ ys) :: Bool
, (x:xs) ++ ys == x:(xs ++ ys) :: Bool
, (xs ++ ys) ++ zs == xs ++ (ys ++ zs) :: Bool
]
Please see the uSpeculate example in the eg folder for the full code
of speculateAbout
.
uSpeculate has some limitations:
 it sometimes prints redundant equations;
 although it usually runs quickly with less than 6 symbols, runtime is exponential with the number of symbols given, providing it with more than a dozen symbols can make it run for several minutes or hours;
 there is no way to configure the size limit of reported equations;
 it only supports variables of
Int
,Bool
,[Int]
, and[Bool]
types.
Please see Speculate for a fullfeatured version without the above limitations.
Example 5: uConjure
Using Express, it takes less than 70 lines of code to define a function
conjure
that generates a function from a partial function definition
and a list of primitives.
Example 5.1. Given:
factorial :: Int > Int
factorial 0 = 1
factorial 1 = 1
factorial 2 = 2
factorial 3 = 6
factorial 4 = 24
Running:
conjure "factorial" factorial
[ val (0 :: Int)
, val (1 :: Int)
, value "+" ((+) :: Int > Int > Int)
, value "*" ((*) :: Int > Int > Int)
, value "foldr" (foldr :: (Int > Int > Int) > Int > [Int] > Int)
, value "enumFromTo" (enumFromTo :: Int > Int > [Int])
]
Prints:
factorial :: Int > Int
factorial x = foldr (*) 1 (enumFromTo 1 x)
Example 5.2. Given:
(+++) :: [Int] > [Int] > [Int]
[x] +++ [y] = [x,y]
[x,y] +++ [z,w] = [x,y,z,w]
Running:
conjure "++" (+++)
[ val (0 :: Int)
, val (1 :: Int)
, val ([] :: [Int])
, value "head" (head :: [Int] > Int)
, value "tail" (tail :: [Int] > [Int])
, value ":" ((:) :: Int > [Int] > [Int])
, value "foldr" (foldr :: (Int > [Int] > [Int]) > [Int] > [Int] > [Int])
]
Prints:
(++) :: [Int] > [Int] > [Int]
xs ++ ys = foldr (:) ys xs
Please see the uConjure example in the eg folder for the full code.
uConjure has some limitations:
 the maximum function size (7) or number of tests (60) are not configurable;
 the maximum function size has to be kept small (<=7) for a reasonable runtime. Due to this, several simple functions are simply outofreach;
 the number of primitive functions given has to be kept small (<12) for a reasonable runtime;
 there is no support for explicitly recursive functions
thought it is possible to pass
foldr
and similar functions as primitives.
Please see Conjure library for an experimental version that addresses some the above limitations.
Further reading
For a detailed documentation, please see Express’s Haddock documentation.
For more examples, see the eg and bench folders.
Express is subject to a paper in the Haskell Symposium 2021 titled “Express: Applications of Dynamically Typed Haskell Expressions”.
Changes
Changelog for Express
v1.0.10

show functionencoded Ordering case expressions exceptionally

show functionencoded Bool case expressions exceptionally

add
caseBool
andcaseOrdering
toData.Express.Fixtures

minor updates in Makefile and CI scripts
v1.0.8

Data.Express.Express.Derive
: fix generation of:
and>:
in earlier GHC’s. 
Data.Express.Utils.TH
: addunboundVars
,toBounded
andtoBoundedQ
.
v1.0.6

fix pretty printing of unapplied infixed variable functions: use
f :: ...
instead of(`f`) :: ...

Data.Express.Fixtures
: addinit'
,div'
,mod'
,quot'
,rem'
,question
andoo
. 
minor fixes in README
v1.0.4
 deeply encode
Ratio
s  add
Express (Complex a)
instance  add several missing
Name
instances deriveName
now usesx
forNum
instances
v1.0.2

more Express instances:
Double
&Float
Int*
types fromData.Int
Word*
types fromData.Word
GeneralCategory
fromData.Char

minor fix in README
v1.0.0
This release indicates that the Data.Express
API is now stable.
 no changes since v0.2.0 or v0.1.16.
v0.2.0
This release indicates that the Data.Express
API is stable.
 no changes since v0.1.16
v0.1.16

add
five
,six
, …twelve
toData.Express.Fixtures
. 
add
cs_
toData.Express.Fixtures
. 
improve backwards compatibility:
Data.Express.Core/Hole/Match/Map/Name/Triexpr/Utils
now work on Hugs. 
100% Haddock coverage on most modules including REPL examples.
v0.1.14

permit and prettyprint
[<n>..<m>]
notations. 
improve default variable names when canonicalizing
 lists are named xs, ys, xss, yss, etc.
 functions are named f, g, h
 before they were simply x, y, z
v0.1.12

Data.Express.Fixtures
, add several symbols:hh
andhhE
;four
andzzs
;signum'
andsignumE
;compose
and.
;mapE
andmap'
.

Add the experimental
Triexpr
module, including: the
Triexpr
type;  tests;
 benchmarks.
 the

Retire Travis as the CI
v0.1.10
 add the
hasHole
andisComplete
functions  add the
encompasses
function  add
appendInt
toData.Express.Fixtures
 add the
uconjure
example  the
Express
typeclass now requiresShow
 improve examples in the
eg/
folder  improve tests of
hasInstanceOf
andisInstanceOf
 improve tests
 add this changelog
v0.1.8
 slightly change behaviour of
canonicalVariations
and related functions.  add more fixtures and improve fixtures’ documentation
 improve Makefile and test scripts
 use GitHub actions as CI
v0.1.6
 add
compareLexicographically
andcompareQuickly
 define behaviour of
canonicalVariations
for some undefined cases  improve haddock documentation
 improve tests
v0.1.4
 add the
fill
andisFun
functions Data.Express.Fixtures
: more fixtures, define fixity add fixity for some fixtures
 improve documentation, tests and lint
v0.1.3
See the git commit log for v0.1.3 and previous versions.